On Sobolev bilinear forms and polynomial solutions of second-order differential equations

Given a linear second-order differential operator $${\mathcal {L}}\equiv \phi \,D^2+\psi \,D$$ L ≡ ϕ D 2 + ψ D with non zero polynomial coefficients of degree at most 2, a sequence of real numbers $$\lambda _n$$ λ n , $$n\geqslant 0$$ n ⩾ 0 , and a Sobolev bilinear form $$\begin{aligned} {\mathcal {B}}(p,q)\,=\,\sum _{k=0}^N\left\langle {{\mathbf {u}}_k,\,p^{(k)}\,q^{(k)}}\right\rangle , \quad N\geqslant 0, \end{aligned}$$ B ( p , q ) = ∑ k = 0 N u k , p ( k ) q ( k ) , N ⩾ 0 , where $${\mathbf {u}}_k$$ u k , $$0\leqslant k \leqslant N$$ 0 ⩽ k ⩽ N , are linear functionals defined on polynomials, we study the orthogonality of the polynomial solutions of the differential equation $${\mathcal {L}}[y]=\lambda _n\,y$$ L [ y ] = λ n y with respect to $${\mathcal {B}}$$ B . We show that such polynomials are orthogonal with respect to $${\mathcal {B}}$$ B if the Pearson equations $$D(\phi \,{\mathbf {u}}_k)=(\psi +k\,\phi ')\,{\mathbf {u}}_k$$ D ( ϕ u k ) = ( ψ + k ϕ ′ ) u k , $$0\leqslant k \leqslant N$$ 0 ⩽ k ⩽ N , are satisfied by the linear functionals in the bilinear form. Moreover, we use our results as a general method to deduce the Sobolev orthogonality for polynomial solutions of differential equations associated with classical orthogonal polynomials with negative integer parameters.

Stable Calculation of Krawtchouk Functions from Triplet Relations

Deployment of the recurrence relation or difference equation to generate discrete classical orthogonal polynomials is vulnerable to error propagation. This issue is addressed for the case of Krawtchouk functions, i.e., the orthonormal basis derived from the Krawtchouk polynomials. An algorithm is proposed for stable determination of these functions. This is achieved by defining proper initial points for the start of the recursions, balancing the order of the direction in which recursions are executed and adaptively restricting the range over which equations are applied. The adaptation is controlled by a user-specified deviation from unit norm. The theoretical background is given, the algorithmic concept is explained and the effect of controlled accuracy is demonstrated by examples.

Generalized coherent states of exceptional Scarf-I potential: Their spatio-temporal and statistical properties

We construct generalized coherent states for the rationally extended Scarf-I potential. Statistical and geometrical properties of these states are investigated. Special emphasis is given to the study of spatio-temporal properties of the coherent states via the quantum carpet structure and the auto-correlation function. Through this study, we aim to find the signature of the “rationalization” of the conventional potentials and the classical orthogonal polynomials.

Robust Stabilization of Interval Plants with Uncertain Time-Delay Using the Value Set Concept

This paper considers the robust stabilization problem for interval plants with parametric uncertainty and uncertain time-delay based on the value set characterization of closed-loop control systems and the zero exclusion principle. Using Kharitonov’s polynomials, it is possible to establish a sufficient condition to guarantee the robust stability property. This condition allows us to solve the control synthesis problem using conditions similar to those established in the loopshaping technique and to parameterize the controllers using stable polynomials constructed from classical orthogonal polynomials.

Dp,q-classical orthogonal polynomials: An algebraic approach

In this paper, we introduce a new technical method for the study of the Dp,q-classical orthogonal polynomials where Dp,q is the (p,q)-difference operator, using basically an algebraic approach. Some new characterizations are given. The approach has been illustrated with three examples.